Question

question given the following unit circle, rotate green dot to the appropriate angle and then find the exact value of the function. \\(\sin 240^{\circ}\\)

Explanation:

Step1: Determine the reference angle

The angle $240^\circ$ is in the third quadrant. To find the reference angle, we subtract $180^\circ$ from $240^\circ$: $240^\circ - 180^\circ = 60^\circ$.

Step2: Recall the sine value of the reference angle

We know that $\sin 60^\circ=\frac{\sqrt{3}}{2}$.

Step3: Determine the sign of the sine value

In the third quadrant, both sine and cosine values are negative (since $x$ and $y$ coordinates on the unit circle are negative there). So, $\sin 240^\circ = -\sin 60^\circ$.

Step4: Substitute the value of $\sin 60^\circ$

Substituting $\sin 60^\circ=\frac{\sqrt{3}}{2}$ into the equation, we get $\sin 240^\circ = -\frac{\sqrt{3}}{2}$.

Answer:

$-\frac{\sqrt{3}}{2}$